Let X0 be the initial state of a Gibbs sampler on an undirected probabilistic graphical model, and XT be the final state. Assume the sampler is initialized in equilibrium, so the distribution of both X0 and XT is the distribution given by the graphical model.
Take any subsets XTR1,...,XTRm of XT, such that the variables in each subset are at least a distance 2T away from the variables in the other subsets (with distance given by shortest path length in the graph). Then XTR1,...,XTRm are all mutually independent given X0.
Let X0 be the initial state of a Gibbs sampler on an undirected probabilistic graphical model, and XT be the final state. Assume the sampler is initialized in equilibrium, so the distribution of both X0 and XT is the distribution given by the graphical model.
Take any subsets XTR1,...,XTRm of XT, such that the variables in each subset are at least a distance 2T away from the variables in the other subsets (with distance given by shortest path length in the graph). Then XTR1,...,XTRm are all mutually independent given X0.